Conformal Invariance of Loops in the Double-Dimer Model

Abstract

The dimer model is the study of random dimer covers (perfect matchings) of a graph. A double-dimer configuration on a graph \({\mathcal{G}}\) is a union of two dimer covers of \({\mathcal{G}}\) . We introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes (relative to a fixed set of faces) of loops in the double dimer model on a planar graph. As an application we prove that, in the scaling limit of the “uniform” double-dimer model on \({\mathbb{Z}^2}\) (or on any other bipartite planar graph conformally approximating \({\mathbb{C}}\)), the loops are conformally invariant.